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This paper proposes a statistical model-based speech dereverberation approach that can cancel the late reverberation of a reverberant speech signal captured by distant microphones without prior knowledge of the room impulse responses. With this approach, the generative model of the captured signal is composed of a source process, which is assumed to be a Gaussian process with a time-varying variance, and an observation process modeled by a delayed linear prediction (DLP). The optimization objective for the dereverberation problem is derived to be the sum of the squared prediction errors normalized by the source variances; hence, this approach is referred to as variance-normalized delayed linear prediction (NDLP). Inheriting the characteristic of DLP, NDLP can robustly estimate an inverse system for late reverberation in the presence of noise without greatly distorting a direct speech signal. In addition, owing to the use of variance normalization, NDLP allows us to improve the dereverberation result especially with relatively short (of the order of a few seconds) observations. Furthermore, NDLP can be implemented in a computationally efficient manner in the time-frequency domain. Experimental results demonstrate the effectiveness and efficiency of the proposed approach in comparison with two existing approaches.